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πislinear if there exists θ∈Rdso that π(x)∈arg maxx∈X/angbracketleftθ, φ(x, a)/angbracketright. For generalized linear contextual bandits, it is clear that the optimal policy must be linear. 2 re-weighted according to the matrix U. In the generalized linear model, the re-weighted regressi on provides a near-optimal con... | https://arxiv.org/abs/2502.13115v1 |
symmetric Gaussian random matrix, i.e., Zij=Zji∼N(0,4σ2 α,β∆2) independently. Joint DP. We first recall the definition of (joint) DP algorithms for non-interactive problems. In this setting, an algorithm maps the dataset D={z1,···, zT}∈ZTto a distribution over the output decision Π. The two dataset D= (z1,···, zT),D′= (z... | https://arxiv.org/abs/2502.13115v1 |
it adopts the following protocol for each round t= 1, ..., T : •Algselects a decision πt∈Π and a ( α, β)-DP channel Qtbased on the history Ht−1={π1, o1,···, πt−1, ot−1}. • The environment generates an observation zt∈Zsampled via zt∼M⋆(πt), where M⋆is the underlying model of the environment. •Algreceives a noisy observa... | https://arxiv.org/abs/2502.13115v1 |
Then (1) For any T-round (α, β)-JDP algorithm Algwith output ˆθ, it holds that sup θ⋆∈[−1,1]Eθ⋆,Alg/bracketleftig Eφ∼p/an}⌊∇a⌋ke⊔le{⊔φ,ˆθ−θ⋆/an}⌊∇a⌋ke⊔∇i}h⊔2/bracketrightig /greaterorsimilarmin/braceleftbigg1 λ(α+β)2T2, λ/bracerightbigg . (2) For any T-round (α, β)-LDP algorithm Algwith output ˆθ, as long as β≤λ T2, ... | https://arxiv.org/abs/2502.13115v1 |
regret bound is meaningful only when the dimension d=O(1) is of constant order. On the other hand, Chen and Rakhlin [2025] provides a significantly improved regret of√ d3T/α. While the Decision-Estimation Coefficient (DEC) approach is much diffe rent from the aforementioned ones, the way they upper bound the DEC implicitly... | https://arxiv.org/abs/2502.13115v1 |
and ( µν/2)-strongly convex over the domain WU:={w∈Rd:/⌊a∇d⌊lUw/⌊a∇d⌊l≤1}. Further, the gradient of LUcan be derived as ∇L U(w) =E(φ,y)∼M⋆/bracketleftbiggUφ /⌊a∇d⌊lUφ/⌊a∇d⌊l(ν(/an}⌊∇a⌋ke⊔le{⊔Uφ, w/an}⌊∇a⌋ke⊔∇i}h⊔)−y)/bracketrightbigg +λLν·Uw. The following lemma indicates that, any approximate minimi zer ofLUis provide... | https://arxiv.org/abs/2502.13115v1 |
log( N),20},Algorithm 1 outputs (U, λ)that satisfies Eq. (2) with probability at least 1−δ, where λ:=λ(K)= (2K+ 5)εN. 4.3 Private regression with reweighting In the following, we present Algorithm 2 for private L1-regression, which is based on (1) first learning the normalization ( U, λ) by the subroutine JDP_Learning _N... | https://arxiv.org/abs/2502.13115v1 |
section, we use our L1-regression method from Section 4 as a subroutine for the action elimination framework [ Li et al. ,2024], providing rate-optimal private regret bounds for genera lized linear contextual bandits. Our algorithm ( Algorithm 3 ) is epoch-based (with a given epoch schedule 1 = T0< T 1< T 2<···< T J=T)... | https://arxiv.org/abs/2502.13115v1 |
Algorithm 12 ) preserves (α, β)-JDP and guarantees that Reg≤˜O/parenleftig dA/radicalbig dTlog|A|+σα,βdAd3/2/parenrightig . In particular, when |A|=O(1), we obtain a upper bound of ˜O/parenleftig√ dT+d3/2 α/parenrightig . Note that the√ dT-term matches the optimal non-private regret bound for |A|=O(1) (up to logari... | https://arxiv.org/abs/2502.13115v1 |
sed methods to provide dimension-free regret bounds in private linear contextual bandits. 3Our approach naturally applies to learning in RKHS. However , to avoid measure-theoretic issues, we only present our algorithms for finite dimensional space. 12 Algorithm 5 JDP_Improper _BatchSGD Input: DatasetD={(φt, yt)}t∈[T]. I... | https://arxiv.org/abs/2502.13115v1 |
n oise ζandφ∼p. Using this idea, we can show that Eq. (9) provides an estimator of ∇L Sq(θ(k)) with small bias for all epochs. Here, the clipping operation is defined as clipR(v):= max{min{v, R},−R}∈[−R, R],∀v∈R. With a careful analysis that bounds the bias introduced by cl ipping (9), we provide the following guarantee... | https://arxiv.org/abs/2502.13115v1 |
D. Pál, and C. Szepesvári. Improved algo rithms for linear stochastic bandits. In Advances in Neural Information Processing Systems , pages 2312–2320, 2011. N. Abe and P. M. Long. Associative reinforcement learning us ing linear probabilistic concepts. In Proceedings of the Sixteenth International Conference on Machine... | https://arxiv.org/abs/2502.13115v1 |
basis for learning. The Journal of Machine Learning Research , 17(1):4062–4095, 2016. J. He, J. Zhang, and R. Q. Zhang. A reduction from linear conte xtual bandits lower bounds to estimations lower bounds. In International Conference on Machine Learning , pages 8660–8677. PMLR, 2022. V. Karwa and S. Vadhan. Finite samp... | https://arxiv.org/abs/2502.13115v1 |
which ensures cλ≤ncand (1−c)Σ−cλI/√∇e⌋edesequal1 nn/summationdisplay i=1Vi/√∇e⌋edesequal(1 + c)Σ + cλI, with probability at least 1 −δ. ✷ Lemma A.4 Suppose that (Xt)tis a vector-valued martingale-difference sequence adapted to the filtration (Ft)t. Assume that/⌊a∇d⌊lXt/⌊a∇d⌊l2≤ctalmost surely. Then it holds that ∀x≥0 P ... | https://arxiv.org/abs/2502.13115v1 |
U(w⋆), we have µν 4/⌊a∇d⌊l/hatwidew⋆ U−w⋆/⌊a∇d⌊l2≤−/an}⌊∇a⌋ke⊔le{⊔∇L U(w⋆),/hatwidew⋆ U−w⋆/an}⌊∇a⌋ke⊔∇i}h⊔≤µνλ/⌊a∇d⌊l/hatwidew⋆ U−w⋆/⌊a∇d⌊l, and hence/⌊a∇d⌊l/hatwidew⋆ U−w⋆/⌊a∇d⌊l≤4λ. Next, for any vector v∈Rd, we have Eφ∼p[|/an}⌊∇a⌋ke⊔le{⊔φ, Uv/an}⌊∇a⌋ke⊔∇i}h⊔|]2≤Eφ∼p[/⌊a∇d⌊lUφ/⌊a∇d⌊l]·Eφ∼p/bracketleftbigg/an}⌊∇a⌋ke⊔l... | https://arxiv.org/abs/2502.13115v1 |
not actually compute (H(K),Σ(K)), and they only appear in our analysis (where we can regard E(K)= 0). Proposition C.3 Suppose that the sequence of matrices {(U(k), H (k),Σ(k))}is defined recursively by Eq. (12) -(14), with/⌊a∇d⌊lE(k)/⌊a∇d⌊lop≤ε. Suppose that λ(k)= (2k+ 5)ε, and ε≤0.1. Then, for any k≥1, it holds that λm... | https://arxiv.org/abs/2502.13115v1 |
, except that we apply Lemma A.3 . Lemma C.4 Suppose that the sequence {(U(k),/tildewideH(k))}is generated by Algorithm 1 . For each k= 0,1,···, K− 1, we define H⋆ (k):=Eφ∼p/bracketleftbiggU1/2 (k)φφ⊤U1/2 (k) /⌊a∇d⌊lU(k)φ/⌊a∇d⌊l/bracketrightbigg . Then, for any fixed parameter c >1, with probability at least 1−δ, the fol... | https://arxiv.org/abs/2502.13115v1 |
recall that /hatwidew⋆ U= arg minw∈WLU(w). In particular, when we take K= 16κνlog(T), we have Lν·/⌊a∇d⌊lw(K)−/hatwidew⋆ U/⌊a∇d⌊l ≤C1/bracketleftigg κ3/2 ν/radicalbigg log(T) log(log( T)/δ) T+σα,β(κ3/2 ν+κν√ d)log(T)/radicalbig log(log( T)/δ) T/bracketrightigg =:ǫ(T, δ).(17) 26 Algorithm 9 Subroutine LDP_Learning _Nor... | https://arxiv.org/abs/2502.13115v1 |
Proposition C.7 andProposition C.8 . Then, Algorithm 8 preserves (α, β)-LDP, and the following holds with probability at least 1−2δ: (1) The normalization (U, λ)satisfies Eq. (2) , and the estimator /hatwidewsatisfies Lν/⌊a∇d⌊l/hatwidew−/hatwidew⋆ U/⌊a∇d⌊l≤ǫ, where λ=λ(T, δ) =˜O/parenleftigg σα,β/radicalbigg dlog(1 /δ) ... | https://arxiv.org/abs/2502.13115v1 |
with σ2≤O/parenleftigσ2 α,β N/parenrightig . Using Lemma A.1 andLemma A.4 , we also have max k/⌊a∇d⌊lZ(k)/⌊a∇d⌊l2≤O/parenleftigdσ2 α,βlog(K/δ) N/parenrightig with probability at least 1 −δ. Applying Proposition C.11 gives the desired upper bounds. ✷ C.6.1 Proof of Proposition C.11 By definition, we have w⋆= ProjW(w⋆... | https://arxiv.org/abs/2502.13115v1 |
b(j)overRd. 31 Theorem D.2 (Meta regret guarantee) Under Assumption 2 ,Algorithm 11 ensures that Reg≤E J−2/summationdisplay j=04dA·N(j+1)Eπ(j)/bracketleftig ˆb(j)(x, a)/bracketrightig + 2N(0)+ 2T Jδ+ 2T2δ0, where N(j)=T(j+1)−T(j)is the batch size of the jth epoch. Further, if the subroutine Regression _with_Confid... | https://arxiv.org/abs/2502.13115v1 |
taking summation over j= 0,1,···, J−2 gives J−1/summationdisplay j=1N(j)·Reg(π(j))≤4dAJ−2/summationdisplay j=0N(j+1)·Eπ(j)/bracketleftig ˆb(j)(x, a)/bracketrightig + 2T2δ0. Note that the above inequality holds under the success event ofAssumption 2 , which holds with probability at least 1−Jδ. Then taking expectation... | https://arxiv.org/abs/2502.13115v1 |
probability at least 1 −δ 6: E2:1 NT/summationdisplay t=N+1U1/2φtφ⊤ tU1/2 /⌊a∇d⌊lUφt/⌊a∇d⌊l+1 2λI/{ollowsequal1 2Eφ∼pU1/2φφ⊤U1/2 /⌊a∇d⌊lUφ/⌊a∇d⌊l. 35 In the following, we condition on E1∩E2. UnderE2, we have D:=1 NT/summationdisplay t=N+1Uφtφ⊤ tU /⌊a∇d⌊lUφt/⌊a∇d⌊l+λU/{ollowsequal1 4I. Note that Ξ = DU−1. For any vector... | https://arxiv.org/abs/2502.13115v1 |
Proofs from Section 6 E.1 Proof of Theorem 6.2 Privacy guarantee. To make the presentation clear, we re-write the iteration of Algorithm 5 as follows. For any sub-dataset Dsub={(φt, yt)}, we denote g(θ;Dsub):=1 |Dsub|/summationdisplay (φ,y)∈D subφ(/an}⌊∇a⌋ke⊔le{⊔φ, θ/an}⌊∇a⌋ke⊔∇i}h⊔−y), F(θ;Dsub):= ProjBd(R)(θ−ηg(θ;Dsu... | https://arxiv.org/abs/2502.13115v1 |
need to prove Eq. (23) for the case i=k. We denote θ+ (k+1)=θ(k)−ηg(k). Using ¯g(θ) =Σ(θ−θ⋆)andEq. (23) recursively for i < k , we know θ+ (k+1)−θ⋆= (I−ηΣ)k+1(θ(0)−θ⋆)−ηk/summationdisplay i=0(I−ηΣ)k−ierr (i). Therefore, /vextenddouble/vextenddoubleθ+ (k+1)/vextenddouble/vextenddouble≤/vextenddouble/vextenddouble/parenl... | https://arxiv.org/abs/2502.13115v1 |
taken over the sequence ζ= (ζ(0),···, ζ(K))of independent Gaussian random vectors. Therefore, we have Eζ/bracketleftigg Eφ∼pexp/parenleftigg |/an}⌊∇a⌋ke⊔le{⊔E(k) 2, φ/an}⌊∇a⌋ke⊔∇i}h⊔|2 4Kσ2 N/parenrightigg/bracketrightigg ≤2,∀k. Therefore, by Markov’s inequality and taking the union boun d, we have Pζ/parenleftigg... | https://arxiv.org/abs/2502.13115v1 |
the privacy guarantee of Algorithm 13 can be implied by the privacy guarantee of the regression oracle (similar to Algorithm 11 ). Lemma E.8 If the oracle Regression preserves (α, β)-JDP (or correspondingly (α, β)-LDP), then Algo- rithm 13 preserves (α, β)-JDP (or correspondingly (α, β)-LDP) Proof of Theorem 6.4 (1). F... | https://arxiv.org/abs/2502.13115v1 |
Arbitrage-free catastrophe reinsurance valuation for compound dynamic contagion claims Jiwook Jang∗Patrick J. Laub†Tak Kuen Siu‡Hongbiao Zhao§ February 20, 2025 Abstract In this paper, we consider catastrophe stop-loss reinsurance valuation for a reinsur- ance company with dynamic contagion claims. To deal with convent... | https://arxiv.org/abs/2502.13325v1 |
events will likely range between US $20 billion and US $30 billion. These events present significant challenges to the financial stability of insurers and reinsurers, highlighting the need for improved models to predict claims from catastrophic events. In May 2021 a ransomware cyberattack halted the US Colonial Pipelin... | https://arxiv.org/abs/2502.13325v1 |
stop-loss reinsurance contracts. We arrange the paper as follows. In Section 2, we provide a mathematical definition for the compound DCP for Ct, which was introduced by Jang and Oh (2021). The infinitesimal generator of the joint process is also provided extending it to the time-inhomogeneous case, together with requi... | https://arxiv.org/abs/2502.13325v1 |
λt=a+ (λ0−a) e−δt+X i≥1Xie−δ(t−T1,i)I{T1,i≤t}+X j≥1Yje−δ(t−T2,j)I{T2,j≤t}, (2.1) where •the initial intensity at time t= 0,λ0>0,P-a.s.; •the constant mean-reverting level a≥0; •the constant rate of exponential decay δ >0. In Definition 2.1, {Xi}i=1,2,...is interpreted as a sequence of positive externally-excited jumps,... | https://arxiv.org/abs/2502.13325v1 |
by ˆh(s) =Z∞ 0e−sxdH(x),ˆg(s) =Z∞ 0e−sydG(y),ˆj(s) =Z∞ 0e−sξdJ(ξ), (2.3) which are supposed to be finite. The Laplace transforms will be used in section 3. The following proposition for a standard CDCP with time-homogeneous parameters is directly adapted from Dassios and Zhao (2011, 2017) and Jang and Oh (2021). 5 Prop... | https://arxiv.org/abs/2502.13325v1 |
dH(x;t)−f(λ, n, c, m, Λ, t) <∞. Setting Af(λ,Λ, n, c, t ) =λin Eq. (2.7), we have Aλ=−κ(t)λ+ρ(t)µH(t) +a(t)δ. Asλt−λ0−Rt 0Aλsdsis anFt-martingale, we have E λt−Zt 0Aλsds|λ0! =λ0. Hence, E[λt|λ0] =λ0−Zt 0κ(s)E[λs|λ0] ds+Zt 0{ρ(s)µH(s) +a(s)δ}ds. Differentiating it with respect to t, we then have the ODE for µλ(t) :=E(λt... | https://arxiv.org/abs/2502.13325v1 |
a reinsurance strategy starting at time t, say{ϕu|u∈[t, T]} ∈ H t, is said to be an arbitrage reinsurance strategy if the respective insurer’s final gain at time Tsatisfies Gt,T(ϕ)≥0,P-a.s. and E[Gt,T(ϕ)|Ft]>0, whereE[·|Ft]is the conditional expectation given Ftunder the probability measure P. Intuitively, an arbitrage... | https://arxiv.org/abs/2502.13325v1 |
E[eℏXt] =E f(Xt)×eℏXt E[eℏXt], (3.2) provided that E eℏXt <∞. Indeed, Eq. (3.2) follows by a version of the Bayes’ rule and setting the Radon–Nikodym derivative for the measures change as the Esscher transform with the Esscher parameter ℏ. From Definition 3.4, to determine an equivalent martingale probability meas... | https://arxiv.org/abs/2502.13325v1 |
0 . (3.5) Also, df1(B) dB|B=0=δ−θˆj(ν)µG>0. (3.6) Taking the second-order derivative of f1(B) with respect to Bgives: d2f1(B) dB2=−θˆj(ν)Z∞ 0euyy2dG(u)<0. Then f1(B) is strictly concave. This, together with Eqs. (3.5) and (3.6), imply that f1(B)> 0, for B∈(0, B+), where B+>0 is the smallest positive solution to f1(B) =... | https://arxiv.org/abs/2502.13325v1 |
∂t+ K′(t) +B′(t)λ ˜f+δ(a−λ) ∂˜f ∂λ+B(t)˜f! +λ ∂˜f ∂Λ+ϕ˜f! +λ θˆj(ν)ˆg(−B(t))Z∞ 0Z∞ 0˜f(λ+y, n+ 1, c+ξ, m, Λ, t)eB(t)y ˆg(−B(t))dG(y)e−νξ ˆj(ν)dJ(ξ)−˜f +ρ" ψˆh(−B(t))Z∞ 0˜f(λ+x, n, c, m + 1,Λ, t)eB(t)x ˆh(−B(t))dH(x)−˜f# . Define d˜G(y;t) :=eB(t)y ˆg(−B(t))dG(y), d˜H(x;t) :=eB(t)x ˆh(−B(t))dH(x), d˜J(ξ) :=e−νξ ˆj(ν)... | https://arxiv.org/abs/2502.13325v1 |
+ ˜ρ(t) Z∞ 0˜f ˜λ+v, n, c, m + 1,˜Λ, t˜h v θˆj(ν)ˆg(−B(t));t θˆj(ν)ˆg(−B(t))dv−˜f . Then, we have the transforms. Note that the parameters of a CDCP under the original measure Pare all constants, whereas the parameters of a CDCP under the new measure P∗become time-varying (except for the intensity-decay rate δ)... | https://arxiv.org/abs/2502.13325v1 |
Then, we have the transforms a→θ 1 +ν γ−ηβ β−B(t)a; ρ→ψα α−B(t)ρ; J∼Gamma (γ, η)→Gamma (γ+ν, η); H∼Exp(α)→Exp(λh); G∼Exp(β)→Exp(λg); where λh:=α−B(t) θ 1 +ν γ−ηβ β−B(t), λ g:=β−B(t) θ 1 +ν γ−ηβ β−B(t), ν ∈(−γ,0). Proof. In particular for exponential distributions, H∼Exp(α), G∼Exp(β), i.e., h(x) =αe−αx, g (y) =βe−... | https://arxiv.org/abs/2502.13325v1 |
be denoted by bE∗[(Ct−L)+], and the corresponding estimate underPwill be denoted by bE[(Ct−L)+]. For this section, we consider α= 2, β= 1, η= 3, γ= 0.4, δ= 3, ρ= 4, a= 1, λ0= 1, t= 1, ν=−0.05, θ= 1.25, ψ= 1.25, ϕ=−(θˆj(ν)−1). For each combination of parameters, 10,000 crude Monte Carlo (CMC) samples of Ctunder P∗have b... | https://arxiv.org/abs/2502.13325v1 |
2 with those in Table 3, the increases in gross insurance and reinsurance premium estimates due to changes in θare greater than those resulting from changes in ψ. This is because ˆj(ν) is involved in the self-exciting intensity function λt. Table 4 illustrates the increase in gross insurance and reinsurance premium est... | https://arxiv.org/abs/2502.13325v1 |
of Theoretical and Applied Finance , 20(01):1750003. Duffie, D., Filipovi´ c, D., and Schachermayer, W. (2003). Affine processes and applications in finance. The Annals of Applied Probability , 13(3):984–1053. Duffie, D., Pan, J., and Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions... | https://arxiv.org/abs/2502.13325v1 |
, t∗ N}uniformly in [0 , T] and sort them 5: Initialize T ← ∅ 6: foreach candidate time t∗in{t∗ 1, . . . , t∗ N}do 7: Sample U∼Uniform(0 ,1) 8: ifU < ρ (t∗)/ρmaxthen 9: Append t∗toT 10: return T In the next algorithm we split the intensity into λt=a(t) + (λ0−a(t)) e−δt+X i≥1Xie−δ(t−T1,i)I{T1,i≤t} | {z } =:λ(c) t,the “C... | https://arxiv.org/abs/2502.13325v1 |
Flow-based generative models as iterative algorithms in probability space Yao Xie∗Xiuyuan Cheng† Abstract Generative AI (GenAI) has revolutionized data-driven modeling by enabling the synthesis of high-dimensional data across various applications, including image generation, language model- ing, biomedical signal proce... | https://arxiv.org/abs/2502.13394v1 |
models can be cast as particle-based iterative algorithms in probability space using the Wasserstein metric, providing both theoretical guarantees and computational efficiency. Based on this frame- work, we establish the convergence of the iterative algorithm and show the generative guarantee, ensuring that under suita... | https://arxiv.org/abs/2502.13394v1 |
t=Ttot= 0 to transport from qto a distribution 3 31SDE trajectoryODE trajectoryFigure 3: Trajectory of SDE versus ODE: The SDE trajectory corresponds to a diffusion model, while for ODE, the dynamics are deterministic, but the initial position of the trajectory follows a distribution. close to p. Normalizing flow model... | https://arxiv.org/abs/2502.13394v1 |
specific, NFs are transformations that map an easy-to-sample initial distribution, such as a Gaussian, to a more complex target distribution. Generally, an NF model provides an invertible flow mapping Fθ:Rd→Rd, parametrized by θ(usually a neural network), such that it maps from the “code” or “noise” z(typically Gaussia... | https://arxiv.org/abs/2502.13394v1 |
xn−17→xnto be invertible. The computation of the inverse mapping, however, may not be direct. After training, the generation ofxis by sampling z∼qand computing z7→xis via the flow map computed through Nblocks. Meanwhile, for the discrete-time flow (11), the computation of the likelihood (9) calls for the computation of... | https://arxiv.org/abs/2502.13394v1 |
use convolutional neural networks (CNN) if the data are images, and graph neural networks (GNN) to generate data on graphs. 3.4 Discrete-time flow as iterative steps We first would like to point out that the distinction between discrete-time versus continuous-time flow models is not strict since continuous-time flow ne... | https://arxiv.org/abs/2502.13394v1 |
a computational bottleneck for high dimensional data is the computation of ∇·vθin (13). The back- propagation training still needs to track the gradient field along the numerical solution trajectories of the neural ODE, which makes the approach “simulation-dependent” and computationally costly. In contrast, the recent ... | https://arxiv.org/abs/2502.13394v1 |
space. We will detail on the flow motivated by the JKO scheme and the theoretical analysis of the generation accuracy. 4.1 Iterative flow using JKO scheme We consider discrete-time flow models where each block can potentially take form of a continuous- time NF (neural ODE), following the set up in Section 3.4. As has b... | https://arxiv.org/abs/2502.13394v1 |
the velocity field vθ(x, t) over the interval t∈[t0, t1] is modeled by a neural network with parameters θ. The empirical version of (22)(23) gives the training objective of the first block as min θ1 mmX i=1xi(t1)2 2−Zt1 t0∇ ·vθ(xi(τ;θ), τ)dτ +1 2γmmX i=1∥xi(t1)−xi(t0)∥2, (24) where γ >0 controls the step size. After ... | https://arxiv.org/abs/2502.13394v1 |
significantly. In contrast, Local FM decomposes this transport into smaller, incremental steps, interpolating between distributions that are closer to each other, hence the name “local.” The Local FM model trains a sequence of small, invertible sub-flow models, which, when concatenated, transform between the data and n... | https://arxiv.org/abs/2502.13394v1 |
(DPI) for f-divergences. Let pandqbe two probability distribution, such that pis absolutely continuous with respect to q. For a convex function f: [0,∞)→Rsuch that fis finite for all x >0,f(1) = 0, and fis right-continuous at 0, the f-divergence of pfrom qis defined as Df(p∥q) =R f p(x) q(x) q(x)dx; for instance, KL-... | https://arxiv.org/abs/2502.13394v1 |
perceptual feature space, which is important in applications where sample quality is crucial, such as image generation. Negative Log-Likelihood (NLL) is a direct measure of how well a generative model fits the training data distribution. Given a test dataset {˜xi}m′ i=1and a model (e.g., learned by flow model) with pro... | https://arxiv.org/abs/2502.13394v1 |
for instance, over a segment [ t0, t1]: Zt1 t0Ex(t)∼ρt∥v(x(t), t)∥2 2dt≈1 (t1−t0)mmX i=1∥xi(t1)−xi(t0)∥2 2. Finally, the KL divergence terms can be estimated from particles at the two endpoints by estimating the log-likelihood function log q(x)/p(x), which can be estimated using a technique training a classification ne... | https://arxiv.org/abs/2502.13394v1 |
is instead induced by a problem-specific risk function R(·) : Rd→R. The formulation is based on the following so-called flow-based DRO problem [41], where the objective is to solve a transport map: min FEX∼p[R(F(X)) +1 2γ∥X−F(X)∥2 2]. (29) 19 for a given regularization parameter γ >0. Here, the reference measure pcan b... | https://arxiv.org/abs/2502.13394v1 |
MPS-MODL-00814643). The authors thank Chen Xu for help with numerical examples. References [1] Michael S Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic in- terpolants. In ICLR , 2023. [2] David Alvarez-Melis, Yair Schiff, and Youssef Mroueh. Optimizing functionals on the space of probabilit... | https://arxiv.org/abs/2502.13394v1 |
current methods. IEEE Transactions on Pattern Analysis and Machine Intelligence , 43(11):3964–3979, 2020. [21] Daniel Kuhn, Peyman Mohajerin Esfahani, Viet Anh Nguyen, and Soroosh Shafieezadeh- Abadeh. Wasserstein distributionally robust optimization: Theory and applications in machine learning. In Operations research ... | https://arxiv.org/abs/2502.13394v1 |
and Yao Xie. Computing high-dimensional optimal transport by flow neural networks. International Conference on Artificial Intelligence and Statistics (AISTATS) , 2025. [41] Chen Xu, Jonghyeok Lee, Xiuyuan Cheng, and Yao Xie. Flow-based distributionally robust optimization. IEEE Journal on Selected Areas in Information ... | https://arxiv.org/abs/2502.13394v1 |
arXiv:2502.13486v1 [eess.SY] 19 Feb 2025KERNEL MEAN EMBEDDING TOPOLOGY: WEAK AND STRONG FORMS FOR STOCHASTIC KERNELS AND IMPLICATIONS FOR MODEL LEARNING BYNACISALDI1,aAND SERDAR YÜKSEL2,b 1Department of Mathematics, Bilkent University,anaci.saldi@bilkent.edu.tr 2Department of Mathematics and Statistics, Queen’s Univers... | https://arxiv.org/abs/2502.13486v1 |
do not require pointwise convergence, which may be too demanding (especially in an em pirical learning theoretic context). A review will be provided later in the paper. 1.1. Summary of Main Results. In this paper, we propose a mathematical framework for kernel mean embedding topologies on stochastic kernels viewed as B... | https://arxiv.org/abs/2502.13486v1 |
[ 12,1,49] offer a comprehensive analysis of various implications of w∗- topology in stochastic control theory, such as the continui ty of expected costs in control policies [ 12], approximation results [ 49] under different cost criteria, and the continuity of invariant measures of diffusions in control policies [ 1].... | https://arxiv.org/abs/2502.13486v1 |
e. Specifically, the unit ball of L∞(µ,M(U))=L1(µ,C0(U))∗is compact under the w∗-topology, resulting in a relatively 4 compact metric topology on stochastic kernels. It is import ant to note that the presentations in [12, Section 3] and [ 2, Section 2.4] differ slightly, although the induced topolo gy remains identical.... | https://arxiv.org/abs/2502.13486v1 |
background (se e Theorem 3.10). THEOREM 1.1. Let{γλ}be a sequence of stochastic kernels from one Borel space to an - other (locally compact) Borel space and let γalso be a stochastic kernel. Then, the following are equivalent: (i)γλ⇀∗γwith respect to kernel mean embedding topology. (ii)γλ⇀∗γwith respect to w∗-topology.... | https://arxiv.org/abs/2502.13486v1 |
replaced by an L2-space because the conditional expectation of an output fun ction belongs to thisL2-space, not the input RKHS. This shift leads to an operator no rm topology for stochastic kernels. Notably, the strong kernel mean embedd ing topology dominates the op- erator norm topology [ 42, Theorem 3.6]. Alternativ... | https://arxiv.org/abs/2502.13486v1 |
Unless otherwise specified, the term ‘measurable’ will refer to Borel measurability. 2. Bochner Spaces and Their Duals. Let(Y,Y,µ)be a probability space, where Yis a Borel space. Let (B,/bardbl·/bardblB)be a separable Banach space and let B∗denote the topological dual ofBwith the induced norm /bardbl·/bardblB∗which turn... | https://arxiv.org/abs/2502.13486v1 |
µ,B∗/parenrightbig ∋γ/ma√sto→/integraldisplay Y/a\}bracketle{tγ(y),f(y)/a\}bracketri}htµ(dy)∈R is continuous for all f∈Lp/parenleftbig µ,B/parenrightbig. We write γλ⇀∗γ, ifγλconverges to γinLq/parenleftbig µ,B∗/parenrightbigwith respect to weak∗-topology. SupposeGis a subset of B∗and define Lq/parenleftbig µ,G/parenrigh... | https://arxiv.org/abs/2502.13486v1 |
and continuous. (b)k(·,u)∈C0(U)for allu∈U. (c)Hkis dense in C0(U). Under above assumptions, any finite signed measure ν∈M(U)can be embedded into Hk as follows: I:M(U)∋ν/ma√sto→Iν/defines/integraldisplay Uk(·,u)ν(du)∈Hk, KERNEL MEAN EMBEDDING TOPOLOGIES FOR STOCHASTIC KERNELS 9 where the integral on the right is in Bochn... | https://arxiv.org/abs/2502.13486v1 |
{u}, we have ν({u})=0 . This means that for any n, ifνnis the empirical estimate of ν, then the total variation distance /bardblνn−ν/bardblTVwill always be 2. Consequently, total variation distance is not suitable fo r estab- lishing empirical consistency. However, when considering weak convergence topology, the strong... | https://arxiv.org/abs/2502.13486v1 |
of weakly∗measurable functions fromYtoHk. Letγ∈Γ. Then, by Lemma 3.7,γis a weakly∗measurable function from YtoM(U) whose range is contained in P(U). Hence, for any g∈Hk⊂C0(U), the following mapping Y∋y/ma√sto→/a\}bracketle{tγ(y),g/a\}bracketri}ht=/integraldisplay Ug(u)γ(y)(du)=/a\}bracketle{tI◦γ(y),g/a\}bracketri}htHk∈... | https://arxiv.org/abs/2502.13486v1 |
TOPOLOGIES FOR STOCHASTIC KERNELS 13 we also have sup g∈Hk:/bardblg/bardbl≤1|Ly(g)|≤1. Hence,Lyis a bounded linear functional on Hk, whereHkis endowed with sup-norm. Since Hkis dense in C0(U), by continuous linear extension theorem, we can uniquely ex tendLyto C0(U)as a bounded linear functional, denoted as ˆLy. But si... | https://arxiv.org/abs/2502.13486v1 |
L2(µ,Hk)is given by: /a\}bracketle{t/a\}bracketle{tγ,f/a\}bracketri}ht/a\}bracketri}ht=/integraldisplay Y/a\}bracketle{tγ(y),f(y)/a\}bracketri}htHkµ(dy). By Theorem 2.1, we know that the topological dual of L2(µ,Hk)is isometrically isomorphic to itself: L2(µ,Hk)∗=L2(µ,Hk). This duality is expressed through the inner pr... | https://arxiv.org/abs/2502.13486v1 |
a locally compact Borel space endowed with its Borel σ-algebraU. For any g∈C0(U), let /bardblg/bardbl/definessup u∈U|g(u)| which turns (C0(U),/bardbl·/bardbl)into a separable Banach space. Let /bardbl·/bardblTVdenote the total variation norm on M(U). The following result is a famous Riesz-Markov-Kakutani re presentatio... | https://arxiv.org/abs/2502.13486v1 |
with respect to w∗-topology (see [ 51]) and is a bounded subset, it is (sequentially) compact and metrizabl e with respect to w∗-topology by Banach-Alaoglu Theorem. Note that Lq/parenleftbig µ,P(U)/parenrightbig is a subset of Lq/parenleftbig µ,P≤1(U)/parenrightbig , and so, we can endow Lq/parenleftbig µ,P(U)/parenrig... | https://arxiv.org/abs/2502.13486v1 |
over the product space, the weak convergence topology is defined as the Young narrow topology. Alternativ ely, this topology can also be de- fined by viewing stochastic kernels as mappings from YtoP(U)and employing duality with the set of Cb-Caratheodory function valued mappings, which is similar t o the construction inw... | https://arxiv.org/abs/2502.13486v1 |
original formulation, the topology is induced by the set of Cb-Caratheodory functions that are in L1(µ,Cb(U)). Here, we extend this definition to Lp(µ,Cb(U))for any1≤p<∞. 20 3.3. Comparison of Weak Topologies. Based on the results established earlier, we ob- serve some intriguing characteristics of the three weak top ol... | https://arxiv.org/abs/2502.13486v1 |
topology), for all i=1,...,k . Since we can approximate any function in Lp(µ,Hk)via a sequence of simple functions pointwise, we can conclude that above convergenc e is true for any f∈Lp(µ,Hk), which concludes the proof of the converse part. Now using the above characterizations of the topologies, we complete the proof... | https://arxiv.org/abs/2502.13486v1 |
that subs equence{νnk}of{νn}also converges with respect to the weak topology. Hence, Gis relatively (sequentially) compact with respect to the weak topology. By Prohorov’s theorem [ 30, Theorem 1.4.12], Gis tight. This completes the proof of item (iii). The proofs of items (i) and (ii) are similar, and so, we only gi v... | https://arxiv.org/abs/2502.13486v1 |
mild assumptions. Their mathematical structure is well- suited for demonstrating such existence results, as weak to pologies simplify the handling of compactness and continuity properties in infinite-dimensi onal spaces. Consequently, it is both natural and effective to define weak topologies on the set of p olicies. For... | https://arxiv.org/abs/2502.13486v1 |
the existence of opti mal solutions in decision problems. 24 However, a trade-off will occur in view of continuity at the e xpense of compactness: In the context of learning and robustness, where uncertainty in th e system dynamics is a central concern, the strong kernel mean embedding topology proves t o be the approp... | https://arxiv.org/abs/2502.13486v1 |
embedding topology on the set of allq-Bochner integrable functions from YtoHk. However, if the range of the function is contained in HP, the situation may differ. The following result demonstrat es that this is not the case. THEOREM 4.1. Letγλ⇀∗γinLq(µ,HP). Then, it is not necessarily the case that γλ→ γin strong kerne... | https://arxiv.org/abs/2502.13486v1 |
we aim to show that the mappi ngsγλ:Y→HP are relatively sequentially compact under the topology of u niform convergence on compacta. To achieve this, two requirements must be met: (i) The set {γλ:Y→HP}should be equicontinuous. (ii) For any y∈Y, the set{γλ(y)(·)}⊂HPmust be relatively compact. Since the MMD topology is e... | https://arxiv.org/abs/2502.13486v1 |
optimal p olicy for discounted cost (see, e.g., [ 29,22]). Hence, in the remainder of this section, we only focus on d eterministic stationary policies. We impose the assumptions below on the components of the Mark ov decision process. ASSUMPTION 2. (a) The one-stage cost function cis inCb(X×A). (b)Xisσ-compact locally... | https://arxiv.org/abs/2502.13486v1 |
that µref(C)=1 and sup (x,a)∈C/bardblpn(·|x,a)−p(·|x,a)/bardblM→0. KERNEL MEAN EMBEDDING TOPOLOGIES FOR STOCHASTIC KERNELS 29 SinceAis finite, one can take the set Cin the following form D×A, whereκabs(D)=1 . Hence, sup (x,a)∈D×A/bardblpn(·|x,a)−p(·|x,a)/bardblM→0. Note that for any (x,a)∈X×A, sincepn(·|x,a)≪κabsandκabs... | https://arxiv.org/abs/2502.13486v1 |
l; that is,l is bounded and continuous, l(·,z)∈C0(Z)for allz∈Z, andHlis dense in C0(Z). For some L>0, define B/defines{f∈Hl:/bardblf/bardblHk≤L}. If{fn}⊂B, then{fn}is equicontinuous and uniformly bounded. To see this, fix any (z,y)∈ Z. Then we have |f(z)−f(y)|=|/a\}bracketle{tf,l(·,z)/a\}bracketri}htHl−/a\}bracketle{tf,l... | https://arxiv.org/abs/2502.13486v1 |
Hence,pθalso converges to pcontinuously in MMD topology (or equivalently in weak con- vergence topology). Therefore, by part (a) of Theorem 5.1, we have |J∗(κ0,pθ)−J∗(κ0,p)|→0. This is contradicting with |J∗(κ0,pλ)−J∗(κ0,p)|>0∀λ. This completes the proof. 5.1.3. Implications for Asymptotic Optimality under Empirical Mo... | https://arxiv.org/abs/2502.13486v1 |
ALDER , E. J. (1997). Consequences of denseness of Dirac Young meas ures. Journal of Mathematical Analysis and Applications 207536–540. [6] B ÄUERLE , N. and L ANGE , D. (2018). Optimal control of partially observable piecew ise deterministic Markov processes. SIAM Journal on Control and Optimization 561441–1462. [7] B... | https://arxiv.org/abs/2502.13486v1 |
A., K ARRAY , F. and C ROWLEY , M. (2021). Reproducing Kernel Hilbert Space, Mercer’s Theorem, Eigenfunctions, Nystr \" om Method, and Use of Kernels in Machine Learning: Tutorial and Survey. arXiv preprint arXiv:2106.08443 . [27] G IHMAN , I. I. and S KOROHOD , A. V. (2012). Controlled Stochastic Processes . Springer ... | https://arxiv.org/abs/2502.13486v1 |
(2017). Kernel Mean Em- bedding of Distributions: A Review and Beyond. Foundations and Trends in Machine Learning 10 1-141. [46] P APAGEORGIOU , N. S. and W INKERT , P. (2018). Applied Nonlinear Functional Analysis: An Introduction . De Gruyter. [47] P ARK, J. and M UANDET , K. (2020). A Measure-Theoretic Approach to K... | https://arxiv.org/abs/2502.13486v1 |
and L ANCKRIET , G. R. G. (2010). Hilbert Space Embeddings and Metrics on Probabilit y Measures. Journal of Machine Learning Research 111517-1561. [61] T AMÁS , A. and C SÁJI, B. C. (2024). Recursive Estimation of Conditional Kernel M ean Embeddings. Journal of Machine Learning Research 251–35. [62] T AMÁS , A. and C S... | https://arxiv.org/abs/2502.13486v1 |
arXiv:2502.13537v1 [math.ST] 19 Feb 2025Precise quantile function estimation from the characteris tic function Gero Junike∗ February 20, 2025 Abstract We provide theoretical error bounds for the accurate numeri cal computation of the quantile function given the characteristic function of a continuous random variable. W... | https://arxiv.org/abs/2502.13537v1 |
Wang (2017). A direct link between ϕandF−1via non-linear integro-differential equations is given in Shaw and McCabe (2009). Suppose His a numerical approximation of F, in the sense that sup y∈R|F(y)−H(y)| ≤ε, (2) given some predefined error tolerance ε >0. Depending on the exact Fourier technique and the numerical integr... | https://arxiv.org/abs/2502.13537v1 |
≤2ε h(y+c)+ε+o(ε). (4) Proof. We use |F−1(p)−H−1 Num(p)| ≤ |F−1(p)−H−1(p)|+|H−1(p)−H−1 Num(p)|. (5) The second term at the right-hand side of Inequality (5) is le ss or equal than εby assumption, which also implies that there is a c∈[−ε,ε] withH−1(p) =y+c. By Assumption A1, it holds thath(y+c)>0. We analyze the first te... | https://arxiv.org/abs/2502.13537v1 |
follows that supy∈R|HCOS(y)−F(y)| ≤ε. Proof. The inequality |HCOS(y)−F(y)|< εfor ally∈Rfollows as in Junike and Pankrashkin (2022, Corollary 9) using Markov’s inequality and the fact t hatfhas semi-heavy tails. In the following Remarks, we provide more details on how to im plement Theorems 2.1 and 2.3. Remark 2.4.In pr... | https://arxiv.org/abs/2502.13537v1 |
We setν=γ= 1 andθ= 0. Then X3has support on Rand has mean 0, variance 1, skewness 0 and kurtosis 6, i.e., much heavier tails than the normal dist ribution. Remark 3.1.On the computation of reference values: We compute referenc e values for Table 1 for Fby the COS method using ( a,b) as in Theorem 2.3 with ε= 10−9. We s... | https://arxiv.org/abs/2502.13537v1 |
RHS of Inequality (4) is (approximately) eq ual to2ε1 min{h1(y1±ε1)}+ε1= 0.73>δ. So,ε1is too large. In the next step, we set ε2:= 0.0005 ≈δ 2 min {h1(y1±ε1)}+1and obtain a(ε2) =−7.9, b(ε2) = 7.9 andN(ε2) = 114. We observe in Table 1 that the RHS of Inequality (4) is n ow satisfied, and Theorem 2.3 ensures that |H−1 Num ... | https://arxiv.org/abs/2502.13537v1 |
An Efficient Permutation-Based Kernel Two-Sample Test Antoine Chatalic1,2, Marco Letizia1,3, Nicolas Schreuder1,4, and Lorenzo Rosasco1,5,6 1MaLGa Center - DIBRIS, Universit `a di Genova, Genoa, Italy 2CNRS, Univ. Grenoble-Alpes, GIPSA-lab, France 3INFN - Sezione di Genova, Genoa, Italy 4CNRS, Laboratoire d’Informatiqu... | https://arxiv.org/abs/2502.13570v2 |
. . . . . . . . . . . . . . 9 1arXiv:2502.13570v2 [stat.ML] 20 Mar 2025 4.2.2 Bound on the empirical quantile threshold . . . . . . . . . . . . . . . . . . . . . 11 5 Numerical studies 11 5.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2 Results . . . . . . ... | https://arxiv.org/abs/2502.13570v2 |
for comparing probability distributions PandQwithout requiring prior assumptions. However, they are often hindered by their computational cost, which scales quadratically with the total sample size nBnX+nY. In order to mitigate this computational drawback, several approximations of the maximum mean discrepancy (MMD) ha... | https://arxiv.org/abs/2502.13570v2 |
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